Trigonometry  Geometry  side lengths
posted by Sam .
In convex quadrilateral ABCD, angle A is congruent to angle C, AB=CD=180, and AD is not equal to BC. The perimeter of ABCD is 640. Find cos A.
Hmm...I drew the diagram, and labeled some things, I drew an altitude from D to AB and did some other things, but no success was obtained. Any help? Thanks

Trigonometry  Geometry  side lengths 
Reiny
joint BD, and use the cosine law twice, once for each triangle
we know AB = DC = 180
let angle A be Ø
let AD = a and BC = b
BD^2 = a^2 + 180^2  2(180)a cosØ
BD^2 = b^2 + 180^2  2(180)bcosØ
a^2 + 180^2  360a cosØ = b^2 + 180^2  360b cosØ
a^2  b^2 = 360a cosØ  360b cosØ
(a+b)(ab) = 360cosØ (ab)
a+b = 360cosØ
cosØ = (a+b)/360
but perimeter = 640
a+b+180+180=640
a+b = 280
then cosØ = 280/360= 7/9 
Trigonometry  Geometry  side lengths 
Sam
ohhhhh that makes sense i should have thought of that thanks!
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